Between Hahn and Euclid

The symmetric Euclidean metric on note-classes corresponds to rms error
when evaluating temperaments; the Hahn metric, it turns out,
corresponds to minimax error. This suggests we might want to look at
metrics between Euclid and Hahn, corresponding to p-th roots of sums
of p-th powers of absolute values. What we get is this:

||3^a 5^b 7^c||_p =
(|a|^p + |b|^p + |c|^p + |b+c|^p + |a+c|^p + |a+b|^p + |a+b+c|^p)^(1/p)

||(a,b,c)||_2 using this definition is twice what I've been using, but
that changes nothing. If we put p=1 into it, I wonder if that is
useful for anything? Paul liked the L1 error; this would be the
corresponding norm on note classes.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

If we put p=1 into it, I wonder if that is
> useful for anything? Paul liked the L1 error; this would be the
> corresponding norm on note classes.

I looked at the p=1 norm around the unison. The first two shells
correspond to the first two shells of Euclidean, leading to this
19-note scale once again:

[1, 21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24,
3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21]

The next shell, with 36 elements, is the union of the third and fourth
Euclidean shells, and the one after that, with 24 elements, is the
Euclidean fifth shell. Beyond this point the shells don't correspond.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> If we put p=1 into it, I wonder if that is
> > useful for anything? Paul liked the L1 error; this would be the
> > corresponding norm on note classes.
>
> I looked at the p=1 norm around the unison. The first two shells
> correspond to the first two shells of Euclidean, leading to this
> 19-note scale once again:
>
> [1, 21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24,
> 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21]
>
> The next shell, with 36 elements, is the union of the third and fourth
> Euclidean shells, and the one after that, with 24 elements, is the
> Euclidean fifth shell. Beyond this point the shells don't correspond.

Cool. What if you center around a deep hole? Thanks.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> If we put p=1 into it, I wonder if that is
> > useful for anything? Paul liked the L1 error; this would be the
> > corresponding norm on note classes.
>
> I looked at the p=1 norm around the unison.

Does this lead to shells with cardinalities = rhombic dodecahedral
numbers?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > If we put p=1 into it, I wonder if that is
> > > useful for anything? Paul liked the L1 error; this would be the
> > > corresponding norm on note classes.
> >
> > I looked at the p=1 norm around the unison.
>
> Does this lead to shells with cardinalities = rhombic dodecahedral
> numbers?

Apparently the latter are customarily defined with respect to the
cubic, not FCC, lattice :(

http://mathworld.wolfram.com/RhombicDodecahedralNumber.html

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Apparently the latter are customarily defined with respect to the
> cubic, not FCC, lattice :(
>
> http://mathworld.wolfram.com/RhombicDodecahedralNumber.html

The handbook doesn't seem to know the sequence 1,13,19,55,79...